Determine the roots of the given cubic equation: $q^{3}-1 q^{2}-25 q+25=0$

Step 1 ：We are given the cubic equation \(q^{3}-1 q^{2}-25 q+25=0\).

Step 2 ：We need to find the roots of this equation. The roots are the values of \(q\) for which the equation becomes zero.

Step 3 ：Using a mathematical software, we find that the roots of the equation are -5, 5, and 1.

Step 4 ：Final Answer: The roots of the given cubic equation are \(\boxed{-5}\), \(\boxed{5}\), and \(\boxed{1}\).